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Abstract

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg-de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhge˘ım-Klibanov method.

Abstract

We consider the problem of active reduction of the structural vibrations induced by the sloshing of large masses of fuel inside a partly full tank. The proposed study focuses on an experimental device mimicking an aircraft wing made of an aluminum rectangular plate equipped with piezoelectric patches at the clamped end and with a cylindrical tip-tank, more or less filled with liquid. After deriving a representative finite dimensional model of the complete system, containing the first $5$ structural modes of the plate and the first $3$ liquid sloshing modes, a controller is computed. Since our main scope is to control the most energetic mode of the structure, a full state feedback method coupled with an observer is used. Finally, the controller is also tested for different initial conditions/perturbations and the results are compared with the ones obtained with an $H_\infty$ controller. Experimental results illustrate the relevance of the chosen strategy.

Abstract

We consider the problem of the active reduction of structural vibrations of a plane wing induced by the sloshing of large masses of fuel inside partly full tank. This study focuses on an experimental device composed of an aluminum rectangular plate equipped with piezoelectric patches at the clamped end and with a cylindrical tip-tank, more or less filled with liquid at the opposite free end. The control is performed through piezoelectric actuators and the main difficulty comes from the complex coupling between the flexible modes of the wing and the sloshing modes of the fuel. First, a partial derivative equation model and then a finite - dimensional approximation, calculated using the first 5 structural modes of the plate and the first 2 liquid sloshing modes, is established. Second, after a model matching procedure, a simultaneous H-infinity control problem associated to the vibration attenuation problem for two different fillings of the tank is stated. Due to the large scale of the synthesis model and to the simultaneous performance requirements, a reduced-order H-infinity controller is computed with HIFOO 2.0 package and is compared with individual designs for different filling levels. Experimental results are finally provided illustrating the relevance of the chosen strategy.

Résumé

Le problème du contrôle actif des vibrations induites par le ballottement du liquide dans un réservoir partiellement rempli est considéré. L'étude proposée ici, est concentrée sur l'analyse d'un dispositif expérimental imitant une aile d'avion réelle, composé d'une plaque rectangulaire en aluminium équipée d'actionneurs et de capteurs piézoélectriques fixées près du coté fixe de ladite plaque et d'un réservoir cylindrique fixé près du coté libre de la plaque. Dans un premier temps, un modèle dimension finie du système, contenant le 5 premiers modes de vibration de la plaque et le 3 premiers modes de ballottement du liquide, est construit. Puis, en utilisant une méthode de recalage, la réponse fréquentielle est améliorée. Finalement, en utilisant la méthode par placement des pôles, un retour d'état couplé avec un observateur de type Luenberger est calculé. Des expérimentations sont ensuite réalisées pour une déformation initiale de la plaque selon le premier mode de flexion. Puis, le comportement du système sur une bande de fréquences plus large est aussi analysé.

Abstract

In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.

Abstract

The goal of this paper is to provide an analysis of the "toolkit" method used in the numerical approximation of the time-dependent Schrödinger equation. The "toolkit" method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.

Mots-Clés / Keywords

Abstract

We are interested in an inverse problem for the wave equation with potential on a starshaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.

Abstract

It is by now well-known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that trying to fit the discrete fluxes, given by discrete approximations of the wave equation, with the continuous one, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform stability imply convergence. In our case, the most difficult part of the analysis is the one corresponding to the uniform stability, that we shall prove using new uniform discrete Carleman estimates, where uniform means with respect to the discretization parameter. We shall then deduce a convergence result for the discrete inverse problems. Our analysis will be restricted to the 1-d case for space semi-discrete wave equations discretized on a uniform mesh using a finite differences approach.

Abstract

We are interested in the inverse problem of the determination of the potential $p(x),~x\in\Omega\subset\mathbb{R}^n$ from the measurement of the normal derivative $\partial_\nu u$ on a suitable part $\Gamma_0$ of the boundary of $\Omega$, where $u$ is the solution of the wave equation $\partial_{tt}u(x,t)-\Delta u(x,t)+p(x)u(x,t)=0$ set in $\Omega\times(0,T)$ and given Dirichlet boundary data. More precisely, we will prove local uniqueness and stability for this inverse problem and the main tool will be a global Carleman estimate, result also interesting by itself.

Abstract

The problem of active reduction of the structural vibrations induced by the sloshing of large masses of fuel inside partly full tank is considered. The proposed study focuses on an experimental device mimicking an aircraft wing made of an aluminum rectangular plate equipped with piezoelectric patches at the clamped end and with a cylindrical tip-tank, more or less filled with liquid. First, a representative finite dimensional model of the complete system containing the first 5 structural modes of the plate and the first 2 liquid sloshing modes is derived via a model matching procedure. A simultaneous H control problem associated to the vibration attenuation problem for two different fillings of the tank is stated. Due to the large scale of the synthesis model and to the simultaneous performance requirements, a reduced-order H controller is computed with HIFOO 2.0 and is compared with individual designs for different filling levels. Experimental results are finally provided illustrating the relevance of the chosen strategy.